When the action is not complex-valued, and has no sign problem, a normalizing flow is guaranteed to exist [1]}, and for \(N>1\) , is far from unique. Normalizing flows have been successfully applied in lattice field theory to accelerate the MCMC sampling [2]}, [3]}. The case of a complex-valued action (a “complex normalizing flow”) is discussed in [4]}: in short, such a normalizing flow yields a perfect contour given by the image of \(\mathbb {R}^N\)  in \(\mathbb {C}^N\)  under the map \(\phi \) . Thus, for the complex normalizing flow to exist, a perfect contour has to exist. The converse also holds: when a perfect contour exists, a normalizing flow (generally far from unique) is also guaranteed to exist.
